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Atoms

Stochastic Electrodynamics: Lessons from Regularizing the Harmonic Oscillator

Theo M. Nieuwenhuizen

Published Jun 10, 2019• 6 citations
Publisher

Abstract

In this paper, the harmonic oscillator problem in Stochastic Electrodynamics is revisited. Using the exact shape of the Lorentz damping term prevents run-away effects. After introducing a cut-off in the stochastic power spectrum and regularizing the stochastic force, all relevant integrals are dominated by resonance effects only and results are derived that stem from those in the quantum ground state. For an orbit with specific position and momentum at an initial time, the average energy and the average rate of energy change are evaluated, which stem with each other. Resonance effects are highlighted along the way. An outlook on the hydrogen ground state problem is provided.

DOI
10.3390/atoms7020059

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In this paper, the harmonic oscillator problem in Stochastic Electrodynamics is revisited. Using the exact shape of the Lorentz damping term prevents run-away effects. After introducing a cut-off in the stochastic power spectrum and regularizing the stochastic force, all relevant integrals are dominated by resonance effects only and results are derived that stem from those in the quantum ground state. For an orbit with specific position and momentum at an initial time, the average energy and the average rate of energy change are evaluated, which stem with each other. Resonance effects are highlighted along the way. An outlook on the hydrogen ground state problem is provided.
The hydrogen ground state problem is a touchstone for the theory of Stochastic Electrodynamics. Recently, we have shown numerically and theoretically that the H-atom self-ionizes after a characteristic time. In another approach we reconsidered the harmonic oscillator and renormalized the stochastic force in order to suppress high-frequency tails so that all frequency integrals are dominated by the physical resonances. In the present work we consider the regularization of the noise in the hydrogen ground state problem. Several renormalization schemes are considered. Some are well behaved, while in other ones, the high frequency renormalization induces pathologies at low frequencies. In no situation we find a way to escape from the previously signaled self-ionization.
Stochastic electrodynamics is a classical theory which assumes that the physical vacuum consists of classical stochastic fields with average energy 1 2 ℏ ω in each mode, i.e., the zero-point Planck spectrum. While this classical theory explains many quantum phenomena related to harmonic oscillator problems, hard results on nonlinear systems are still lacking. In this work the hydrogen ground state is studied by numerically solving the Abraham–Lorentz equation in the dipole approximation. First the stochastic Gaussian field is represented by a sum over Gaussian frequency components, next the dynamics is solved numerically using OpenCL. The approach improves on work by Cole and Zou 2003 by treating the full 3 d problem and reaching longer simulation times. The results are compared with a conjecture for the ground state phase space density. Though short time results suggest a trend towards confirmation, in all attempted modellings the atom ionises at longer times.
We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first-order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.
The Klein Gordon equation for the harmonic oscillator may be written as (E-.5kx2)2=p2+mo2 ((1)) where mo is the rest mass and p the relativistic momentum. In the literature, it is argued that the resulting x4 term dominates for large x and prevents the Klein Gordon equation from being solved for bound states. As a result, it is replaced with an equation of the form: (E2-mo2) = (p-imwx)(p+imw) ((2)) where p=-ihbar d/dx in one dimension which may be solved exactly. Recently, the authors of (1) have used perturbation theory to calculate energy changes to ((1)) by using -.5V2 - e2 +2eV as a perturbative potential where V=.5kx2 and E=e+mo. They utilise the unperturbed wavefunction exp(-ax2) so large x4 values are damped. Their results are close to the exact calculations of ((2)) for the case of k=1 and m=1. In this note, we argue that for an oscillator there are classical turning points at which the quantum average kinetic energy -1/2m d/dx d/dx W should be zero and small nearby even if the average kinetic energy near x=0 corresponds to relativistic momenta. Thus, it is argued there are different spatial regions, each with its own quantum equation. The Schrodinger equation may be applied to regions near the classical turning point moving to infinite. In such a case, there is no issue with an x4 term. Near x=0, the full Klein Gordon equation ((1)) may be applied. There are questions of then matching the two sets of solutions. In this note, we attempt to investigate this scenario and compare results with those of (1) and (2). (1) and (2) suggest similar relativistic corrections to the nonrelativistic ground state oscillator energy of more than 20%. In this note, we suggest a smaller value.
By virtue of stochastic decoupling, the quantum evolution of a harmonic oscillator is converted into two virtual contributions dictated by either a stochastic creation operator or a stochastic annihilation operator. Two approaches are provided to resolve these virtual two evolution operators. One is based on the coherent-state representation and the other deals with the momentum representation after separating out the position and momentum factors. The quantum propagator of the harmonic oscillator is obtained by taking the random average of the combined virtual propagators. The main mathematical exercise of the whole procedure is the calculation of the Gaussian integral.